Giải thích các bước giải:
Ta có:
$A=(\dfrac{1}{2^2}-1)(\dfrac{1}{3^2}-1)...(\dfrac{1}{2012^2}-1)$
$\to A=-(1-\dfrac{1}{2^2})(1-\dfrac{1}{3^2})...(1-\dfrac{1}{2012^2})$
$\to A=-\dfrac{2^2-1}{2^2}.\dfrac{3^2-1}{3^2}...\dfrac{2012^2-1}{2012^2}$
$\to A=-\dfrac{(2-1)(2+1)}{2^2}.\dfrac{(3-1)(3+1)}{3^2}...\dfrac{(2012-1)(2012+1)}{2012^2}$
$\to A=-\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}...\dfrac{2011.2013}{2012^2}$
$\to A=-\dfrac{1.2....2011}{2.3...2012}.\dfrac{3.4...2013}{2.3...2012}$
$\to A=-\dfrac{1}{2012}.\dfrac{2013}{2}$
$\to A=-\dfrac{2013}{4024}$
